Сборник лекций по общей теории относительности(С. Н. Вергелес)
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gij |
eai , i, a = |
1, . . . , n |
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eai gij ebj = ηab, |
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±1 |
ηab = diag(1, −1, . . . , −1),
ηab = diag(1, 1, . . . , 1).
eia
ea = eia ∂x∂ i
ea · eb = ηab.
n
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X, Y, Z T01X
X(Y · Z) ≡ Xi ∂x∂ i (Y · Z) = ( X Y ) · Z + Y · ( X Z).
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U
gAB;i ≡ gAB,i − ωAiC gCB − ωBiC gAC = 0.
Xi gAB,iY AZB + gAB Y,iAZB + Y AZ,iB =
= gABXi Y,iA + ωCiA Y C ZB + Y A Z,iB + ωCiB ZC .
X
gAB,i − ωAiC gCB − ωBiC gAC Y AZB = 0.
YZ
xµ = (x0, xi), i = 1, . . . , (n − 1))
gµν ;λ = gµν ,λ − ρµλgρν − ρνλgµρ = 0.
ηab,i = 0
ωab + ωba = 0, ωab = ηacωbc, ωab = ωcaηcb.
eaµ
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eµ |
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eaµeµb = δab ←→ eaµeνa = δνµ, gµν = ηabeµa eνb ←→ gµν eaµebν = ηab. |
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ωa = eµa d xµ, d s2 = ηabωaωb. |
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µea = ωaµb eb → ∂µeaλ + τλµeaτ = ωaµb ebλ |
eµeb |
= δb |
a µ |
a |
∂µeλa − ωaµb eλb + λτµeτa = 0 ←→ ∂µeaν + ωbµa ebν − λνµeaλ = 0.
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ωa = eµa d xµ |
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xµ |
(xµ + d xµ) |
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X1, . . . , Xn |
Ω = X[11 X22 . . . Xrn] |
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{Xsa} |
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Ω = ω0 ω1 . . . ωn−1 |
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ωa′ = Λaa′ ωa det Λaa′ = ±1 |
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Ω = eµ01 eµ12 . . . eµnn−1 d xµ1 d xµ2 . . . d xµn = |
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= (det eµa ) d x0 . . . d xn−1. |
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g = det g|µν |
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det ea |
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xµ
p p
d V = |g| d x0 d x1 . . . d xn−1 ≡ |g| d(n) x
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{eA} xµ |
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xµ(s) |
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X(s) |
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d xµ |
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( µX) |
A |
d xµ |
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∂µXA |
A |
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= 0. |
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X| l = d s |
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= d s |
+ ωBµX |
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d XA = −ωA (x(s)) d xµ(s)XB = −ωA XB .
Bµ B l
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X(0) |
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X(s) |
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X(0) |
s = 0 |
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XA(0) |
s |
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d xµ(s)/ds = x˙ µ(s) |
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xµ(s) |
l
ls0
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s0 |
s
ls
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X(s)X(s) = 0, |
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X(s) |
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l |
s |
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α 6= 0 β |
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s = αs′ + β |
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s |
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s −→ s′ |
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X(s) |
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s′(s) |
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X(s) −→ X(s |
) = ds/ds |
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XA −→ dxµ/ds |
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d2xµ(s) |
+ νλµ |
(x(s)) |
dxν (s) |
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dxλ(s) |
= 0. |
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ds |
ds |
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xµ(s0) |
x˙ µ(s0) ≡ (d xµ/ d s) (s0) |
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l
xµ(s), 0 ≤ s ≤ 1, xµ(0) = xµ(1).
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xµ(0) |
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X = XAeA |
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xµ(0) |
xµ(1) |
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X′ |
xµ(1) = xµ(0) |
X = X′ − X |
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xµ(0) |
X X′ |
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X = XAeA |
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XA = Il d XA(s) = − Il ωBA(s)XB(s), |
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ωBA(s) = ωBµA (x(s)) d xµ(s). |
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XB (s) |
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X |
xµ(0) |
xµ(s) |
l |
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xµ(0) = 0 |
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xµ = 0 |
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τ = (τ0, . . . , τn−1) |
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(τ0)2 + . . . + (τn−1)2 = ε2. |
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s |
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τ |
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xτµ (s) = τµs, 0 ≤ s ≤ 1. |
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ε |
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xµ = 0 |
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xµ = 0 |
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xτµ (s) |
s |
{eA} |
xµ = 0 |
{e¯A}τ ,s |
ε |
{e¯A} |
e¯A|xµ=0 = 0.
xµ = 0
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{e¯A} |
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xµ = 0 |
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ω¯BA |
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C1 |
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|ω¯BµA (x)| < C1 ε |
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x |
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ε |
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ε |
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(x0)2 + |
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. . . (xn−1)2 |
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C2ε |
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| XA| < Cε2 ||X||0. |
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xµ = 0 |
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||X||0 = |X(0) |
|X(0) |
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XA |
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xµ = 0 |
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{e¯A}τ ,s |
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ε |
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¯ |
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xµ(0) = 0 |
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µ |
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A {e¯A} |
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xτ |
(s) |
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X (0) |
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O(ε2) |
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XA = − Il ωBA(s)X¯ B (s). |
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¯ B |
(s) |
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¯ |
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X¯ |
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X¯ B(s) − XB(s) |
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xτµ |
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xµ(s) |
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xµ = 0 |
¯ B |
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s′ |
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− |
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ε3 |
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X |
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(s) |
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x (0) |
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X = 0 |
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¯ A |
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A ¯ B |
+ ζ |
A |
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d X |
= −ωB X |
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ζA |
ε ε |
xµ(0) |
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σ |
l : ∂σ = l |
I Z Z
− A ¯ B − A ¯ B − A · ¯ B − A ¯ B
ωB X = d(ωB X ) = (d ωB X ωB d X ).
∂σ σ σ
ζA
XA = −1 Z RA XB .
2 σ B
XB
Xxµ(0)
RBA = 2 (dωBA + ωCA ωBC)
{eA′ } |
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eA′ = φAA′ eA |
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RA′′ |
= φA′ |
φB′ RA. |
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B |
A |
B |
B |
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XA |
XB |
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RA |
XB |
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XA |
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(1, 1) |
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(∂/∂xµ) |
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ωBλA d xλ → νλµ d xλ, |
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RBA → Rνµ = 2 d |
νρµ d xρ |
+ σλµ d xλ νρσ |
d xρ |
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= Rνµ λρ d xλ d xρ, |
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ν λρ |
= ∂λ |
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νλ |
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σ . |
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Rµ |
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∂ρ µ |
+ µ σ |
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µ |
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νρ |
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σλ νρ |
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σρ νλ |
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W = {wµ} |
{xµ} |
V = {vµ} |
(εV, εW, −εV, −εW ), ε → 0 |
Xµ |
l |
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O(ε2) |
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Xµ = −12ε2 (Rνµ λρv[λwρ])Xν.
σ |
l |
xµ(s, t) = ε(s vµ + t wµ), 0 ≤ s, t ≤ 1
σ
d xµ = ε (vµ d s + wµ d t),
d xλ d xρ = ε2(vλwρ − vρwλ) d s d t = ε2v[λwρ] d s d t.
xµ(s) |
xµ(0) = xµ(1) |
I d xµ
l d s d s = 0.
d xµ
s(s + d s)
ε |
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xµ = 0 |
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µ |
{eA(x)} |
x |
µ |
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e |
x |
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{ A( B)} |
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= 0 |
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{e˜A(x)} |
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˜ |
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e˜A(x) = ξ(A)(x) eB (0) |
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|xµ| O(ε) |
d xµ = −xµ |
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˜B |
( B) µ |
B |
B |
µ |
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2 |
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ξ |
(A) |
A |
Aµ |
µ 2 |
x |
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x = δ + ω |
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x + O |
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e˜A(x) − eA(0) = ωAµx eB (0) + O (|x | ) xµ = 0 |
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ε → 0 xµ |
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d e˜A = ωAµB d xµ eB (0) + δe˜A. |
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δe˜A |
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xµ = 0 |
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θ |
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θ(X) = X |
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θ = eA ωA.
θ(X) = eA ωA(X) = eAXA = X.
l |
εµ |
xµ = 0 |
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Hl e˜A ωA |
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x = 0 |
σ |
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∂σ = l |
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I∂σ e˜A ωA = Zσ d(˜eAωA) = Zσ(d e˜A ωA + e˜A d ωA). |
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δe˜A |
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O(ε2) |
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Il e˜AωA |
= Zσ(d ωA + ωBA ωB)eA = |
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Zσ T AeA. |
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T A = 2 d ωA + ωBA ωB |
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T A |
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ωλ = dxλ ωνµ = νλµ dxλ |
T µ = 2 νλµ dxλ |
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dx |
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Tνλµ = µλν − µνλ.
µλν = µνλ.
d s2 = gµν d xµ d xν = gAB ωAωB,
d ωA + ωBA ωB = 12TBCA ωB ωC ,
d ωBA + ωCA ωBC = 12RBA CDωC ωD.
gAB ηab
d ◦ d = 0
d ωa d ωba
R[AB CD] = T[ABC; D] + TFA[BTCDF ],
RBA [CD; F ) = −RBA E[F TCDE ].
ζABC
ζ[ABC]
ζ[ABC] = ζABC + ζBCA + ζCAB, A = eµA µ,
ζABC
ζ[ABC]
Rµ[ν λρ] = 0, Rµν [λρ; σ] = 0, Rµν λρ = gµσRσ |
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ν λρ |
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Rab cd = ηaeRbe cd
10.
Rab cd = −Rab dc
Rab cd = −Rba cd
ωc ωd = −ωd ωc
20.
Rab cd = Rcd ab.
(i) = Rab cd + Rac db + Rad bc = 0.